13 MEASURES WITH BASE {JL 179 that the set of classes A, as A runs through £', is dense in I(X, //,). Show that h(u) = h(u9 a) (Kolmogoroff-Sinai theorem). (Same method as in (a).) (d) Take X to be the unit' circle U : |z| = 1 , let ju, be the image of Lebesgue measure under the mapping t\—*e2nlt of [0, 1] onto U, and let u be the mapping zt-+e2niez. Show that h(u) = 0. (Distinguish two cases according as 6 is rational or irrational. In the latter case use (b), by taking a to be a partition into two half-open semicircles.) 11. Let X, Y be two compact spaces and let (JL (resp. v) be a positive measure on X (resp. Y) such that /x(X) = v(Y) = 1. A /^--measurable (resp. ^-measurable) mapping u : X i— > X (resp. v : Y *—»• Y) such that U(JJL) = ju,(resp. v(v) = v) defines an endomorphism U:f\-*(f°u)~ of L£(X, /^) (resp. an endomorphism V:g^(g°v)~ of Lc(Y, v)) (Section 13.11, Problem 10). The mappings u, v are said to be conjugate if there exists an isometry T of I(X, ju) onto I(Y, v) (Problem 8) such that FQ r= T° U. Show that if this is the case then h(u) — /z(y). 12. Suppose that the measure ^ is bounded, and let p e [1, 4- oo[. Let (£/„) be a sequence of continuous linear mappings of L&(X) into the space S(X, p,) (Problem 2). For any function of fe JS?£ we denote by Un-fany function belonging to the class Un "/ and pu4 (tf!^ /)(*)== sup | (%•/)(*) |, For each a > 0 let E«,N(/) denote the set of all x e X such that (US •/)(*) > a, and let E«(/) denote the set of all x e X such that (U* •/)(*) > a. (a) Show that the set of /e L£ such that fi(Ea,N(/)) ^ e is closed, for each e > 0. (b) Assume that for all /e jgfg the function C7* -/is finite almost everywhere. Show that under these conditions the number C(a)= sup /x(E«(/)) Np(/)«l tends to 0 as a -*• -f <x> (Banach's principle). (Use Baire's theorem in the complete space L& .) (c) Under the hypotheses of (b), show that the set H of classes / such that the sequence ((£/„ •/)(>;)) converges everywhere in X is closed in L&. (Write remark that R(/)(*) = R(f—g)(x) almost everywhere for ff e H, and use (b).) 13. MEASURES WITH BASE p (13.13.1) Let g be a mapping ofX into C or into E. Then the following con- ditions are equivalent: (a) For each x e X, there exists a neighborhood Vofx in X such that g<py is integrable. the set of